Topological Feature Classification Based on Prior Knowledge

Abstract

Techniques for topological classification of complex data sets are provided. For example, one or more embodiments described herein may include a system that may include a memory that stores computer-executable components. The system may also include a processor operatively coupled to the memory and may execute the computer-executable components stored in the memory. The computer-executable components may include a quantum computing component that may encode eigenvalues of a Laplacian matrix into phases on quantum states of a quantum circuit. The computer-executable components may also include a classical computing component that uses a Bayesian learning algorithm to infer Betti numbers by measuring ancillary states of the quantum circuit.

Description

Technical Field

[0001] The present disclosure relates to topological feature classification and, more particularly, to prior knowledge-based inference of one or more Betti numbers using a hybrid of quantum and classical computing techniques. Background Art

[0002] Topological data analysis techniques utilize algebraic topology formulas to provide a framework that can be insensitive to a particular selected metric, provide dimensionality reduction, and / or be robust to noise found in the subject data. Additionally, topological data analysis can be facilitated by machine learning, deep learning, classification, inference, and / or artificial intelligence tasks. For example, one or more machine learning tasks can utilize the principles of algebraic topology to generate increasingly high-level abstractions of the subject data, where determinations about the characteristics of the data can then be performed based on the abstractions rather than the raw data.

[0003] Traditionally, classical computing techniques or quantum computing techniques have been implemented to perform topological data analysis. Classical computing techniques can readily perform topological data analysis on large data sets. Given the low cost and availability of the processing power of classical computing devices, classical computing techniques can exhibit desirable efficiency when analyzing large data sets. However, analysis by classical computing techniques may be limited by the complexity of the data. In contrast, quantum computing techniques can readily perform topological data analysis on complex data sets but are limited in terms of the amount of data that can be processed. For example, conventional quantum algorithms may require circuit depths or levels of fault tolerance that cannot be easily achieved in near-term quantum computing.

[0004] Therefore, there is a need in the art to address the above problems. Summary of the Invention

[0005] In a first aspect, the following invention provides a system for topological classification, the system comprising: a memory that stores computer-executable components; a processor operably coupled to the memory and executing the computer-executable components stored in the memory, wherein the computer-executable components include: a quantum computing component operable to encode eigenvalues into the phase of the quantum state of a quantum circuit; and a classical computing component that infers Betti numbers using a Bayesian learning algorithm by measuring the ancilla state of the quantum circuit.

[0006] In another aspect, the following invention provides a computer-implemented method for topological classification, the method comprising: encoding, by a system operably coupled to a processor, eigenvalues into the phase of the quantum state of a quantum circuit; and inferring, by the system, Betti numbers using a Bayesian learning algorithm by measuring the ancilla state of the quantum circuit.

[0007] From another aspect, the present invention provides a computer program product for topological classification. The computer program product includes a computer-readable storage medium that can be read by a processing circuit and stores instructions for a method to be executed by the processing circuit to perform the steps of the present invention.

[0008] From another aspect, the present invention provides a computer program stored on a computer-readable medium and loadable into the internal memory of a digital computer. The computer program includes software code portions that, when the program runs on the computer, are used to perform the steps of the present invention.

[0009] An overview is given below to provide a basic understanding of one or more embodiments of the present invention. This overview is not intended to identify key or important elements, or to delineate any scope of a particular embodiment or any scope of the claims. Its sole purpose is to present concepts in a simplified form as a prelude to the more detailed description that follows. In one or more embodiments described herein, systems, computer-implemented methods, devices, and / or computer program products are described that can facilitate the topological classification of complex data sets.

[0010] According to one embodiment, a system is provided. The system may include a memory that can store computer-executable components. The system may further include a processor operably coupled to the memory and capable of executing the computer-executable components stored in the memory. The computer-executable components may include a quantum computing component that can encode the eigenvalues of a Laplacian matrix into the phases of the quantum states of a quantum circuit. These computer-executable components may also include a classical computing component that infers Betti numbers using a Bayesian learning algorithm by measuring the ancillary states of the quantum circuit. An advantage of such a system may be to utilize quantum computing for complex calculations and classical computing for large-scale calculations.

[0011] In some examples, the Betti numbers may be included within a plurality of Betti numbers inferred by the classical computing component, and the plurality of Betti numbers may characterize the topology of an object. An advantage of such a system may be the topological classification of complex data sets.

[0012] According to one embodiment, a computer-implemented method is provided. The computer-implemented method may include encoding the eigenvalues of a Laplacian matrix into the phases of the quantum states of a quantum circuit by a system operably coupled to a processor. The computer-implemented method may further include the system inferring Betti numbers using a Bayesian learning algorithm by measuring the ancillary states of the quantum circuit. An advantage of such a computer-implemented method may be the autonomous implementation of a deep learning task that can incorporate prior knowledge.

[0013] In some examples, the computer-implemented method may include repeatedly measuring an auxiliary state by the system. The auxiliary state may be a product state with a quantum state. The computer-implemented method may further include generating a probability distribution by the system based on the repeated measurements. Additionally, the computer-implemented method may include analyzing the probability distribution using a Monte Carlo sampling algorithm by the system to update a prior knowledge distribution. An advantage of such a computer-implemented method may be that the analysis of the system can be refined based on previous calculations to improve accuracy and / or efficiency.

[0014] According to one embodiment, a computer program product for topological classification is provided. The computer program product may include a computer-readable storage medium having program instructions implemented thereon, the program instructions being executable by a processor to cause the processor to encode eigenvalues of a Laplacian matrix into phases of quantum states of a quantum circuit by a system operatively coupled to the processor. The program instructions may further cause the processor to infer Betti numbers using a Bayesian learning algorithm by the system by measuring an auxiliary state of the quantum circuit. An advantage of such a computer program product may be using quantum computing to facilitate topological classification without introducing undesirable fault tolerance.

[0015] In some examples, the program instructions may further cause the processor to generate subsample objects by selectively sampling objects by the system. Additionally, the program instructions may cause the processor to characterize the subsample objects using a persistent homology algorithm by the system to generate a quantum superposition state. Additionally, the program instructions may cause the processor to generate a Laplacian matrix as a quantum object based on the quantum superposition state by the system. An advantage of such a program product may be topological classification of complex data sets.

[0016] According to one embodiment, a system is provided. The system may include a memory that may store computer-executable components. The system may further include a processor operatively coupled to the memory and may execute the computer-executable components stored in the memory. The computer-executable components may include a quantum computing component that may encode eigenvalues into phases of quantum states of a quantum circuit. The eigenvalues may be related to topological features of an object. The computer-executable components may further include a classical computing component that may infer Betti numbers using a Bayesian learning algorithm by measuring an auxiliary state of the quantum circuit. An advantage of such a system may be leveraging quantum computing for complex calculations and classical computing for topological classification.

[0017] In some examples, the system can further include a subsampling component that can generate a subsampled object by selectively sampling an object. Additionally, the system can include a homology component that can use a persistent homology algorithm to characterize the subsampled object to produce a quantum superposition state. Further, the system can include a matrix component that can generate a Laplacian matrix as a quantum object based on the quantum superposition state. The advantage of such a system can be to split out data from complex datasets to facilitate topological classification through quantum and / or classical computing techniques.

[0018] According to one embodiment, there is provided a computer program product for topological classification. The computer program product can include a computer-readable storage medium having program instructions implemented thereon, the program instructions being executable by a processor to cause the processor to encode eigenvalues into the phases of the quantum states of a quantum circuit by a system operatively coupled to the processor, wherein the eigenvalues are related to the topological features of an object. These program instructions can also cause the system to infer Betti numbers using a Bayesian learning algorithm by measuring the ancillary state of the quantum circuit. The advantage of such a computer program product can be to use quantum computing to facilitate topological classification with a desired circuit depth.

[0019] In some instances, these program instructions can further cause the processor to generate a Bayesian learning algorithm by the system based on previous measurements of the quantum circuit. The advantage of such a program product can be to incorporate prior knowledge into the topological classification of complex datasets. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] The present invention will now be described, by way of example only, with reference to the preferred embodiments as shown in the following drawings:

[0021] Figure 1 A block diagram of an example non-limiting system including one or more quantum computing components that can facilitate topological feature classification, according to one or more embodiments described herein.

[0022] Figure 2A A diagram of an example non-limiting object that can be subjected to topological data analysis by one or more systems, according to one or more embodiments described herein.

[0023] Figure 2B A diagram of an example non-limiting subsampled object that can be generated by one or more quantum computing components to facilitate topological data analysis, according to one or more embodiments described herein.

[0024] Figure 3 A block diagram of an example non-limiting system including one or more quantum computing components that can facilitate topological feature classification, according to one or more embodiments described herein.

[0025] Figure 4 A diagram showing an example non - restrictive simplex construction that can be generated by one or more quantum computing components according to one or more embodiments described herein to facilitate topological data analysis.

[0026] Figure 5 A block diagram showing an example non - restrictive system including one or more quantum computing components that can facilitate topological feature classification according to one or more embodiments described herein.

[0027] Figure 6 A diagram showing an example non - restrictive quantum circuit according to one or more embodiments described herein, which can encode one or more eigenvalues into one or more quantum phases to facilitate topological data analysis.

[0028] Figure 7 A block diagram showing an example non - restrictive system including one or more quantum computing components and classical computing components that can facilitate topological feature classification according to one or more embodiments described herein.

[0029] Figure 8A A diagram showing an example non - restrictive chart that can depict a probability distribution, which can be generated to facilitate the inference of one or more Betti numbers according to one or more embodiments described herein.

[0030] Figure 8B A diagram showing an example non - restrictive chart that can depict a probability distribution, which can be generated to facilitate the inference of one or more Betti numbers according to one or more embodiments described herein.

[0031] Figure 9 A flowchart showing an example non - restrictive method according to one or more embodiments described herein, which can utilize a hybrid scheme of both quantum computing techniques and classical computing techniques to facilitate topological data analysis.

[0032] Figure 10 A flowchart showing an example non - restrictive method according to one or more embodiments described herein, which can utilize a hybrid scheme of both quantum computing techniques and classical computing techniques to facilitate topological data analysis.

[0033] Figure 11 Depicts a cloud computing environment according to one or more embodiments described herein.

[0034] Figure 12 Depicts an abstract model layer according to one or more embodiments described herein.

[0035] Figure 13A block diagram is shown of an example non - limiting operating environment in which one or more embodiments described herein may be facilitated. Detailed Description

[0036] The following detailed description is merely illustrative and is not intended to limit the embodiments and / or the application or uses of the embodiments. Further, there is no intention to be bound by any express or implied information presented in the preceding background or overview sections or in the detailed description section.

[0037] One or more embodiments will now be described with reference to the drawings, where like reference numerals are used throughout to refer to like elements. In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a more thorough understanding of one or more embodiments. However, it will be apparent that one or more embodiments may be practiced without these specific details in different instances.

[0038] Given the problems of conventional implementations of topological data analysis, the present disclosure may be implemented to use a hybrid of classical computing techniques and quantum computing techniques to produce solutions to one or more of these problems in the form of a priori - knowledge - based inference of Betti numbers. Advantageously, incorporating a priori knowledge may overcome one or more drawbacks regarding the dimensionality of quantum computing. Additionally, the incorporation of classical computing techniques may mitigate one or more challenges associated with fault tolerance in conventional quantum computing techniques. Further, the use of quantum computing techniques may facilitate the analysis of complex data sets, including data sets regarding applications such as: recommendation systems, neutrino detection, analyzing data collected from the Large Hadron Collider (“LHC”), chemical compound reuse, chemical analysis, combinations thereof, and / or the like.

[0039] Various embodiments of the present invention may relate to computer - processing systems, computer - implemented methods, devices, and / or computer - program products that facilitate efficient, effective, and autonomous (e.g., without direct human guidance) topological classification. For example, one or more embodiments may infer one or more Betti numbers by incorporating a priori knowledge in a hybrid of classical computing techniques and quantum computing techniques. For example, one or more quantum - computing components may encode one or more eigenvalues into the phases of quantum states to facilitate the determination of a probability distribution. Additionally, one or more classical - computing components may combine knowledge of the probability distribution to infer one or more Betti numbers via conditional probability (e.g., such as the implementation of a Bayesian learning algorithm).

[0040] The computer processing system, computer-implemented method, apparatus, and / or computer program product employ hardware and / or software to solve problems that are inherently highly technical (e.g., topological classification), not abstract, and cannot be performed as a set of mental acts of a human. For example, the different embodiments described herein may contemplate one or more machine learning tasks and / or may implement quantum computing algorithms to facilitate the analysis of complex data.

[0041] Figure 1 A block diagram of an example non-limiting system 100 that may use a hybrid of classical and quantum computing techniques to facilitate the topological classification of complex data sets is shown. For brevity, the repeated description of similar elements employed in other embodiments described herein is omitted. Aspects of the systems (e.g., system 100, etc.), apparatus, or processes in various embodiments of the present invention may constitute one or more machine-executable components implemented within one or more machines (e.g., implemented in one or more computer-readable media associated with one or more machines). When executed by one or more machines (e.g., computers, computing devices, virtual machines, etc.), such components may cause the machines to perform the described operations.

[0042] As Figure 1 shown, system 100 may include one or more servers 102, one or more networks 104, and / or one or more input devices 106. Server 102 may include quantum computing components 108. Quantum computing components 108 may further include communication components 110 and / or subsampling components 112. Similarly, server 102 may include at least one memory 116 or be otherwise associated therewith. Server 102 may further include a system bus 118 that may be coupled to different components, such as but not limited to quantum computing components 108 and associated components, memory 116, and / or processor 120. Although server 102 is shown in Figure 1 , in other embodiments, multiple different types of devices may be associated with or include the features shown in Figure 1 . Further, server 102 may communicate with one or more cloud computing environments via one or more networks 104. Figure 1

[0043] One or more networks 104 may include wired and wireless networks, including but not limited to cellular networks, wide area networks (WANs) (e.g., the Internet), or local area networks (LANs). For example, server 102 may communicate with one or more input devices 106 (and vice versa) using almost any desired wired or wireless technology, including for example but not limited to: cellular, WAN, Wi-Fi, Wi-Max, WLAN, Bluetooth technology, combinations thereof, and / or the like. Additionally, although in the illustrated embodiment the quantum computing components 108 may be disposed on one or more servers 102, it should be understood that the architecture of system 100 is not limited thereto. For example, the quantum computing components 108 or one or more components of the quantum computing components 108 may be located on another computing device, such as another server device, a client device, etc.

[0044] One or more input devices 106 may include one or more computerized devices, which may include but are not limited to: personal computers, desktop computers, laptop computers, cellular phones (e.g., smart phones), computerized tablet computers (e.g., including a processor), smart watches, keyboards, touchscreens, mice, combinations thereof, and / or the like. A user of system 100 may utilize one or more input devices 106 to input data into system 100, thereby sharing the data with server 102 (e.g., via a direct connection and / or via one or more networks 104). For example, one or more input devices 106 may send data to communication component 110 (e.g., via a direct connection and / or via one or more networks 104). Additionally, one or more input devices 106 may include one or more displays, which may present one or more outputs generated by system 100 to the user. For example, the one or more displays may include but are not limited to: cathode ray tube displays ("CRT"), light emitting diode displays ("LED"), electroluminescent displays ("ELD"), plasma display panels ("PDP"), liquid crystal displays ("LCD"), organic light emitting diode displays ("OLED"), combinations thereof, and / or the like.

[0045] A user of system 100 may input one or more settings and / or commands into system 100 using one or more input devices 106 and / or one or more networks 104. For example, in the embodiments described herein, a user of system 100 may operate and / or manipulate server 102 and / or associated components via one or more input devices 106. Additionally, a user of system 100 may use one or more input devices 106 to display one or more outputs (e.g., displays, data, visualizations, etc.) generated by server 102 and / or associated components. In embodiments, a user of system 100 may use one or more input devices 106 and / or one or more networks 104 to provide one or more complex data sets to server 102. As used herein, the term "complex data set" may refer to one or more data sets that include high richness features for underlying patterns in a subject data set. For example, one or more complex data sets may include data that can be characterized by non-linear patterns. Complex data sets may be generic in nature (e.g., not bound to a specific application) or tailored according to a specific context. Additionally, in one or more embodiments, one or more input devices 106 may be included within and / or operatively coupled to a cloud computing environment. Cloud computing techniques may be used to collect and / or analyze a wide variety of complex data sets.

[0046] Quantum computing component 108 may analyze one or more complex data sets shared by one or more input devices 106 to facilitate topological classification. For example, communication component 110 may receive data (e.g., complex data sets) from one or more input devices 106 (e.g., via a direct electrical connection and / or over one or more networks 104) and share the data with different associated components of quantum computing component 108.

[0047] Sub-sampling component 112 may selectively sample one or more objects characterized by a complex data set (e.g., input into system 100 via one or more input devices 106) to generate one or more sub-sampled objects. Sub-sampling component 112 may reduce the size of the data initially analyzed by quantum computing component 108 through the selective sampling performed by sub-sampling component 112. Example selective sampling techniques that may be performed by sub-sampling component 112 may include, but are not limited to: random grid selection, random scatter selection, point cloud selection, Monte Carlo selection, boosting selection, combinations thereof, and / or the like. For example, sub-sampling component 112 may selectively sample a complex data set via one or more point clouds to generate one or more sub-sampled objects.

[0048] Figure 2AA diagram showing an example non - restrictive graph 200 that can depict an exemplary object 202 according to one or more embodiments described herein. The exemplary object 202 can be analyzed by one or more quantum computing components 108. For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted. The exemplary object 202 can be characterized by one or more data sets (e.g., complex data sets) input into the system 100 via one or more input devices 106. For example, the exemplary object 202 shown in graph 200 can be the number three.

[0049] Figure 2B A diagram showing an example non - restrictive graph 204 that can depict an example sub - sample object 206 generated by one or more sub - sampling components 112 based on the exemplary object 202 shown in graph 200. For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted. For example, one or more sub - sampling components 112 can selectively sample the exemplary object 202 (e.g., characterized by one or more data sets such as complex data sets) to generate one or more example sub - sample objects 206 (e.g., as Figure 2B shown). For example, one or more sub - sampling components 112 can generate the example sub - sample object 206 shown in graph 204 via one or more point cloud techniques.

[0050] Figure 3 A diagram showing an example non - restrictive system 100 according to one or more embodiments described herein that further includes a homology component 302. For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted. One or more homology components 302 can use one or more persistent homology algorithms to characterize one or more sub - sample objects generated by one or more sub - sampling components 112 to generate quantum superposition states.

[0051] For example, the homology component 302 can utilize one or more persistent homology algorithms to discern one or more local features (e.g., clusters and / or holes) of one or more sub - sample objects (e.g., example sub - sample object 206). For example, the homology component 302 can generate a plurality of simplices to depict the connectivity between points selectively sampled from the initially analyzed object (e.g., exemplary object 202). One or more homology components 302 can build one or more simplicial complexes from one or more simplicial complexes sharing connectivity based on a defined distance. In addition, the homology component 302 can generate one or more quantum superposition states to facilitate encoding on a quantum circuit.

[0052] In one or more embodiments, the homology component 302 can generate one or more quantum superposition states (“ρ”) based on Equation 1 presented below.

[0053]

[0054] where "S k " can represent a simplex state, and "|S k ><S k |" can represent the quantum state of the topic simplex.

[0055] Figure 4 FIG. shows a diagram of an exemplary non-limiting simplex construction 400 that can be generated by one or more homology components 302 according to one or more embodiments described herein. For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted. Figure 4 The exemplary simplex construction 400 depicted in Figure 2B can be generated by one or more homology components 302 based on the exemplary sub-sample object 206 depicted in Figure 4 . As shown in

[0056] Figure 5 FIG. shows a diagram of an exemplary non-limiting system 100 that further includes matrix components 502 according to one or more embodiments described herein. For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted. One or more matrix components 502 can generate one or more Laplacian matrices as quantum objects based on the quantum superposition states generated by one or more homology components 302.

[0057] For example, one or more matrix components 502 may generate one or more Laplacian matrices as quantum graph Laplacians, which may encode one or more eigenvalues into one or more phases of a quantum state. One or more eigenvalues may be related to and / or otherwise describe one or more topological features discerned by one or more homology components 302. In embodiments, one or more eigenvalues may characterize general topological features (e.g., the one or more eigenvalues need not be specific to a particular topological feature). Thus, one or more matrix components 502 may generate one or more Laplacian matrices that include eigenvalues related to topological features discerned by data connectivity established by one or more homology components 302. Additionally, one or more matrix components 502 may encode eigenvalues into one or more phases of a quantum state by using quantum gates that may project the absolute value of an expected value onto the phase of a quantum state.

[0058] Figure 6 A diagram showing an example non-limiting quantum circuit 600 that may facilitate the performance of quantum encoding performed by one or more matrix components 502, in accordance with one or more embodiments described herein. For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted.

[0059] As Figure 6 shown, the exemplary quantum circuit 600 may include one or more first quantum states 602 (e.g., represented by ) and one or more ancillary states 604 (e.g., represented by "+"). One or more ancillary states 604 may be product states of one or more first quantum states 602. The one or more ancillary states 604 may include one or more first quantum gates 606, which may incorporate one or more variational parameters into the quantum computation. For example, "σ" may be a variational parameter representing a Bayesian risk parameter, and / or "μ" may be a variational parameter representing the mean of a target distribution in an optimization protocol. Incorporating one or more variational parameters via one or more first quantum gates 606 may facilitate solving one or more topic inference problems that one or more quantum computing components 108 are solving.

[0060] Additionally, an exemplary quantum circuit 600 may include one or more second quantum gates 608 located on one or more first quantum states 602. The one or more second quantum gates 608 may facilitate the quantum encoding described herein with respect to the one or more matrix components 502. For example, the one or more second quantum gates 608 may facilitate encoding one or more eigenvalues into one or more phases on the one or more first quantum states 602. In one or more embodiments, the one or more second quantum gates 608 are not limited to a particular type of quantum gate; rather, one of ordinary skill in the art will recognize that the type of quantum gate that may be applied may depend on the subject analysis and / or the circuit configuration.

[0061] As Figure 6 shown, one or more ancillary states 604 may be measured after the quantum encoding facilitated by the one or more second quantum gates 608. In Figure 6 it is also shown that the measurement of the one or more ancillary states 604 may be performed by a classical computing device. For example, the exemplary quantum circuit 600 depicts the described transition to classical computing by drawing two lines extending from the measurement indication. Thus, a classical computing measurement of the exemplary quantum circuit 600 may be achieved. Repeated operations and / or measurements of the exemplary quantum circuit 600 may enable the generation of one or more probability distributions of one or more encoded eigenvalues (e.g., as depicted by the equation shown in Figure 6 where "E" may represent the result of a classical computing measurement of a quantum state).

[0062] Figure 7 FIG. shows an example non-limiting system 100 further including one or more classical computing components 702 in accordance with one or more embodiments described herein. For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted. In embodiments, the one or more quantum computing components 108 and / or the one or more classical computing components 702 may be operably coupled via direct electrical connections and / or via one or more networks 104. For example, in one or more embodiments, the one or more classical computing components 702 may be operably coupled to the one or more quantum computing components 108 via one or more cloud computing environments. Additionally, in embodiments, the one or more classical computing components 702 may include one or more subsampling components 112 (e.g., the selective sampling described herein may be performed by quantum and / or classical computing techniques).

[0063] One or more classical computing components 702 may include a measurement component 704, a sampling component 706, and / or a Bayesian component 708. One or more measurement components 704 may use one or more classical computing methods to measure one or more ancillary states 604 of the quantum computing component 108. For example, the measurement component 704 may measure one or more ancillary states 604 according to Equation 2 presented below.

[0064]

[0065] where "i" may represent a complex value (e.g., "k" may represent a regularization term, and / or "H" may represent a Hamiltonian operator or a Hermitian operator). In one or more embodiments, one or more measurement components 704 may repeatedly measure one or more ancillary states 604 to generate one or more probability distributions according to Equation 2.

[0066] One or more sampling components 706 may use one or more sampling algorithms to analyze one or more probability distributions generated by one or more measurement components 704 to update one or more prior knowledge distributions. Example sampling algorithms may include but are not limited to: random grid selection, random scatter selection, point cloud selection, Monte Carlo selection, boosting selection, combinations thereof, and / or the like. For example, one or more prior knowledge distributions may characterize previous measurements of one or more ancillary states 604 measured by one or more measurement components 704. Those of ordinary skill in the art will readily recognize that the one or more analyses described herein can be adapted to a wide variety of general prior knowledge distributions.

[0067] One or more Bayesian components 708 may generate one or more Bayesian learning algorithms based on one or more prior knowledge distributions and one or more probability distributions.

[0068] In addition, one or more Bayesian components 708 may use the one or more Bayesian learning algorithms to infer one or more Betti numbers. The one or more Betti numbers may be related to one or more topological features distinguished and / or encoded by one or more quantum computing components 108. Equation 3 below may characterize an exemplary Bayesian learning algorithm that may be generated and / or used by one or more Bayesian components 708 to infer one or more Betti numbers.

[0069]

[0070] where, may represent one or more prior knowledge distributions.

[0071] Accordingly, one or more classical computing components 702 can incorporate prior knowledge into inferring one or more Betti numbers, which can describe topological features of an object (e.g., exemplary object 202) undergoing analysis by system 100 via conditional probability. Additionally, one or more classical computing components 702 can share one or more inferred Betti numbers with one or more quantum computing components 108 to further refine one or more persistent homology algorithms in subsequent analysis iterations performed by the one or more quantum computing components 108 and their associated components. Accordingly, in a feedback loop, the computations implemented by the one or more quantum computing components 108 can form the basis for one or more computations implemented by the one or more classical computing components 702, and vice versa. This feedback loop can be iterated until convergence regarding the inferred Betti numbers and / or until a number of times defined by a user of system 100 (e.g., via one or more input devices 106) is reached. Accordingly, system 100 can classify different topological features of an object characterized by one or more complex data sets through a hybrid of quantum and classical computing, which can incorporate prior knowledge to refine one or more analysis processes.

[0072] Figure 8A FIG. shows an example non - limiting diagram 800 depicting Betti number inference that can be performed by one or more classical computing components 702 according to one or more embodiments described herein. For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted. Diagram 800 can be generated by one or more classical computing components 702 based on Figure 4 the exemplary simplicial construction 400 shown. As shown in FIG. 8, "b0" can represent the number of connected components discerned from the exemplary simplicial construction 400. For example, classical computing components 702 can infer that an object (e.g., exemplary object 202) analyzed by one or more quantum computing components 108 includes 1 connected component.

[0073] Figure 8B FIG. shows a diagram of an example non - limiting diagram 802 depicting another Betti number inference that can be performed by one or more classical computing components 702 according to one or more embodiments described herein. For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted. Diagram 802 can also be generated by one or more classical computing components 702 based on Figure 4 the exemplary simplicial construction 400 shown. As shown in FIG. 8, "b1" can represent the number of holes discerned from the exemplary simplicial construction 400. For example, classical computing components 702 can infer that an object (e.g., exemplary object 202) analyzed by one or more quantum computing components 108 includes 0 holes.

[0074] Figure 9 FIG. 900 is a flow chart of an exemplary non - limiting method that can facilitate one or more topological classifications of complex data sets using a hybrid of quantum and classical computing techniques, in accordance with one or more embodiments described herein. For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted.

[0075] At 902, method 900 may include encoding one or more eigenvalues of one or more Laplacian matrices into one or more phases of a quantum state (e.g., one or more first quantum states 602) of a quantum circuit (e.g., exemplary quantum circuit 600) by system 100 operatively coupled to one or more processors 120. For example, the encoding may be performed according to different features of one or more quantum computing components 108 described herein. For example, the encoding at 902 may include: selectively sampling objects, characterizing sub - objects via one or more persistent homology algorithms, constructing one or more simplicial complexes, and / or generating one or more Laplacian matrices.

[0076] At 904, method 900 may include inferring one or more Betti numbers by system 100 using one or more Bayesian learning algorithms by measuring one or more ancillary states (e.g., exemplary ancillary states 604) of the quantum circuit (e.g., exemplary quantum circuit 600). For example, the inference may be performed according to different features of one or more classical computing components 702 described herein. For example, the inference at 904 may include: measuring one or more ancillary states, generating one or more probability distributions, generating and / or updating one or more prior knowledge distributions, and / or using conditional probability techniques incorporating prior knowledge (e.g., one or more Bayesian learning algorithms) to infer the one or more Betti numbers.

[0077] Figure 10 FIG. 1000 is a flow chart of an exemplary non - limiting method that can facilitate one or more topological classifications of complex data sets using a hybrid of quantum and classical computing techniques, in accordance with one or more embodiments described herein. For brevity, repeated descriptions of similar elements employed in other embodiments described herein are omitted.

[0078] At 1002, method 1000 may include generating one or more subsample objects by system 100 operatively coupled to one or more processors 120 by selectively sampling one or more objects. For example, the generation of the one or more subsample objects may be performed according to different features of one or more subsample components 112 described herein. For example, one or more point clouds may be used to perform the generation at 1002.

[0079] At 1004, method 1000 may include characterizing the one or more sub-sample objects by system 100 using one or more persistent homology algorithms to generate one or more quantum superposition states. For example, the one or more sub-sample objects may be characterized according to different features of one or more homology components 302 described herein. For example, the characterization at 1004 may include generating one or more simplicial constructs to discern the connectivity of the one or more sub-sample objects.

[0080] At 1006, method 1000 may include generating, by system 100, one or more Laplacian matrices as one or more quantum objects based on the one or more quantum superposition states. For example, generating the one or more Laplacian matrices may be performed according to different features of one or more matrix components 502 described herein. For example, the one or more Laplacian matrices may include one or more eigenvalues related to one or more topological features discerned during the characterization at 1004.

[0081] At 1008, method 1000 may include encoding, by system 100, one or more eigenvalues of the one or more Laplacian matrices into one or more phases of one or more quantum states of one or more quantum circuits. For example, encoding the one or more eigenvalues may be performed according to different features of one or more quantum computing components 108 (e.g., via one or more matrix components 502). In various embodiments, the quantum circuit may include one or more ancillary states, which may be product states of one or more quantum states encoded with one or more eigenvalues.

[0082] At 1010, method 1000 may include repeatedly measuring, by system 100, the one or more ancillary states of the quantum circuit. For example, the measurement may be performed according to different features of one or more measurement components 704 described herein. The repeated measurement at 1010 may provide one or more classical computations that describe the subject quantum circuit. For example, at 1012, method 1000 may include generating, by system 100, one or more probability distributions based on the repeated measurement at 1010.

[0083] At 1014, method 1000 may include analyzing, by system 100, the one or more probability distributions using one or more Monte Carlo sampling algorithms to update one or more prior knowledge distributions. For example, the analysis may be performed according to different features of one or more sampling components 706 described herein. In one or more embodiments, the Monte Carlo sampling algorithms may facilitate refining the one or more prior knowledge distributions (e.g., via noise reduction).

[0084] At 1016, method 1000 may include generating, by system 100, one or more Bayesian learning algorithms based on the one or more prior knowledge distributions and / or the one or more probability distributions. For example, generating the one or more Bayesian learning algorithms may be performed according to different features of one or more Bayesian components 708 described herein. For example, the one or more Bayesian learning algorithms generated at 1016 may be characterized by Equation 3.

[0085] At 1018, method 1000 may include inferring, by system 100, one or more Betti numbers using the one or more Bayesian learning algorithms generated at 1016. For example, inferring the one or more Betti numbers may be performed according to different features of one or more classical computing components 702 described herein (e.g., via one or more Bayesian components 708). Thus, the inference of the one or more Betti numbers may be based on one or more measurements of one or more ancillary states of one or more quantum circuits and / or may incorporate one or more prior knowledge distributions to improve accuracy and / or efficiency.

[0086] It should be understood that although this disclosure includes a detailed description of cloud computing, the implementation of the teachings recited herein is not limited to a cloud computing environment. Instead, embodiments of the present invention are capable of being implemented in conjunction with any other type of computing environment now known or later developed.

[0087] Cloud computing is a service delivery model for enabling convenient, on-demand network access to a shared pool of configurable computing resources such as networks, network bandwidth, servers, processing, memory, storage, applications, virtual machines, and services that can be rapidly provisioned and released with minimal management effort or interaction with the service provider. The cloud model can include at least five characteristics, at least three service models, and at least four deployment models.

[0088] The characteristics are as follows:

[0089] On-demand self-service: A cloud consumer can unilaterally and automatically provision computing capabilities such as server time and network storage as needed without human interaction with the service provider.

[0090] Broad network access: The capabilities are available over a network and accessed through standard mechanisms that facilitate use by heterogeneous thin or thick client platforms such as mobile phones, laptop computers, and PDAs.

[0091] Resource pooling: The provider's computing resources are pooled to serve multiple consumers using a multi-tenant model, where different physical and virtual resources are dynamically assigned and reassigned as needed. There is a sense of location independence, as consumers generally have no control or knowledge of the exact location of the provided resources, but may be able to specify a location at a higher level of abstraction (e.g., country, state, or data center).

[0092] Rapid elasticity: Capabilities can be provided quickly and elastically (automatically in some cases) to quickly scale down and quickly release to quickly scale up. To the consumer, the capabilities available for provisioning generally appear unlimited and can be purchased in any quantity at any time.

[0093] Measured service: The cloud system automatically controls and optimizes resource use by leveraging a metering capability at some level of abstraction appropriate to the service type (e.g., storage, processing, bandwidth, and active user accounts). Resource use can be monitored, controlled, and reported, providing transparency for both the provider and consumer of the utilized service.

[0094] The business model is as follows:

[0095] Software as a Service (SaaS): The capability provided to the consumer is to use the provider's applications running on the cloud infrastructure. The applications can be accessed from different client devices via a thin client interface such as a web browser (e.g., web-based email). The consumer does not manage or control the underlying cloud infrastructure, including the network, servers, operating systems, storage, or even individual application capabilities, with the possible exception of limited user-specific application configuration settings.

[0096] Platform as a Service (PaaS): The capability provided to the consumer is to deploy applications created or acquired by the consumer on the cloud infrastructure, where the applications are created using programming languages and tools supported by the provider. The consumer does not manage or control the underlying cloud infrastructure, including the network, servers, operating systems, or storage, but has control over the deployed applications and possibly the application hosting environment configuration.

[0097] Infrastructure as a Service (IaaS): The capability provided to the consumer is to provide processing, storage, networks, and other basic computing resources that the consumer can deploy and run any software that may include operating systems and applications. The consumer does not manage or control the underlying cloud infrastructure, but has control over the operating systems, storage, deployed applications, and possibly limited control over selected networking components (e.g., host firewalls).

[0098] The deployment model is as follows:

[0099] Private cloud: The cloud infrastructure is for the exclusive use of an organization. It can be managed by the organization or a third party and can exist on - premise or off - premise.

[0100] Community cloud: The cloud infrastructure is shared by several organizations and supports a specific community with shared concerns (e.g., tasks, security requirements, policies, and compliance considerations). It can be managed by the organization or a third party and can exist on - premise or off - premise.

[0101] Public cloud: Makes the cloud infrastructure available to the general public or a large industry group and is owned by an organization that sells cloud services.

[0102] Hybrid cloud: The cloud infrastructure is a composition of two or more clouds (private, community, or public) that remain unique entities but are bound together by standardized or proprietary technologies that enable data and application portability (e.g., cloud bursting for load balancing between clouds).

[0103] The cloud computing environment is service - oriented, focusing on statelessness, low coupling, modularity, and semantic interoperability. The core of cloud computing is the infrastructure that includes a network of interconnected nodes.

[0104] Now refer to Figure 11 , which depicts an illustrative cloud computing environment 1100. As shown, the cloud computing environment 1100 includes one or more cloud computing nodes 1102, and local computing devices used by cloud consumers (such as a personal digital assistant (PDA) or cellular phone 1104, desktop computer 1106, laptop computer 1108, and / or in - vehicle computer system 1110) can communicate with the cloud computing nodes 1102. The nodes 1102 can communicate with each other. They can be physically or virtually grouped (not shown) in one or more networks, such as the private cloud, community cloud, public cloud, or hybrid cloud or a combination thereof described above. This allows the cloud computing environment 1100 to provide infrastructure, platform, and / or software as a service such that cloud consumers do not need to maintain resources on local computing devices. It should be understood that Figure 11 the types of computing devices 1104 - 1110 shown in

[0105] Now refer to Figure 12 , which shows a set of functional abstraction layers provided by the cloud computing environment 1100 ( Figure 11 ). For brevity, the repeated description of similar elements employed in other embodiments described herein is omitted. It should be understood in advance that Figure 12The components, layers, and functions shown are illustrative only, and embodiments of the present invention are not limited thereto. As depicted, the following layers and corresponding functions are provided.

[0106] The hardware and software layer 1202 includes hardware and software components. Examples of hardware components include: a host 1204; a server 1206 based on a RISC (Reduced Instruction Set Computer) architecture; a server 1208; a blade server 1210; a storage device 1212; and network and networking components 1214. In some embodiments, the software components include network application server software 1216 and database software 1218.

[0107] The virtualization layer 1220 provides an abstraction layer from which the following examples of virtual entities can be provided: virtual servers 1222; virtual memories 1224; virtual networks 1226, including virtual private networks; virtual applications and operating systems 1228; and virtual clients 1230.

[0108] In one example, the management layer 1232 can provide the functions described below. Resource provisioning 1234 provides for the dynamic acquisition of computing resources and other resources for performing tasks within a cloud computing environment. Metering and pricing 1236 provides cost tracking when resources are utilized within a cloud computing environment and bills or invoices for the consumption of these resources. In one example, these resources can include application software licenses. Security provides authentication for cloud consumers and tasks, as well as protection for data and other resources. The user portal 1238 provides access to the cloud computing environment for consumers and system administrators. Service level management 1240 provides cloud computing resource allocation and management such that the required service levels are met. Service level agreement (SLA) planning and fulfillment 1242 provides for the pre-arrangement and procurement of cloud computing resources based on the expected future requirements of the cloud computing resources according to the SLA.

[0109] The workload layer 1244 provides examples of functions that can utilize the cloud computing environment. Examples of workloads and functions that can be provided from this layer include: mapping and navigation 1246; software development and lifecycle management 1248; virtual classroom education delivery 1250; data analysis processing 1252; transaction processing 1254; and topology classification 1256. Embodiments of the present invention can utilize the cloud computing environment described in Figure 11 and 12 to communicate between one or more components (e.g., between one or more quantum computing components 108, classical computing components 702, and / or input devices 106), and / or facilitate the inference of prior knowledge of Betti numbers from complex data sets.

[0110] The present invention can be a system, method, and / or computer program product at any possible level of integration of technical details. The computer program product may include a computer-readable storage medium (or media) having computer-readable program instructions thereon for causing a processor to execute aspects of the present invention. A computer-readable storage medium can be a tangible device that can retain and store instructions for use by an instruction execution device. A computer-readable storage medium may be, for example but not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing. A non-exhaustive list of more specific examples of computer-readable storage media includes the following: a portable computer disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), a static random access memory (SRAM), a portable compact disc read-only memory (CD-ROM), a digital versatile disc (DVD), a memory stick, a floppy disk, a mechanically encoded device (such as a raised structure or a punched card having instructions recorded thereon), and any suitable combination of the foregoing. As used herein, a computer-readable storage medium should not be construed as a transient signal per se, such as a radio wave or other freely propagating electromagnetic wave, an electromagnetic wave propagating through a waveguide or other transmission medium (e.g., an optical pulse through an optical fiber cable), or an electrical signal transmitted through a wire.

[0111] The computer-readable program instructions described herein can be downloaded from a computer-readable storage medium to a corresponding computing / processing device, or downloaded to an external computer or external storage device via a network (e.g., the Internet, a local area network, a wide area network, and / or a wireless network). The network can include copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface in each computing / processing device receives the computer-readable program instructions from the network and forwards the computer-readable program instructions for storage in a computer-readable storage medium within the corresponding computing / processing device.

[0112] The computer-readable program instructions for performing the operations of the present invention may be assembly instructions, instruction set architecture (ISA) instructions, machine instructions, machine-related instructions, microcode, firmware instructions, state-setting data, configuration data for integrated circuits, or source code or object code written in any combination of one or more programming languages, including object-oriented programming languages such as Smalltalk, C++, etc., and procedural programming languages such as the "C" programming language or similar programming languages. The computer-readable program instructions may be executed entirely on the user's computer, partially on the user's computer, executed as a stand-alone software package, partially on the user's computer and partially on a remote computer, or entirely on the remote computer or server. In the latter case, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or may be connected to an external computer (e.g., through the Internet using an Internet service provider). In some embodiments, an electronic circuit, including, for example, a programmable logic circuit, a field-programmable gate array (FPGA), or a programmable logic array (PLA), may execute the computer-readable program instructions by utilizing the state information of the computer-readable program instructions to personalize the electronic circuit so as to perform aspects of the present invention.

[0113] Aspects of the present invention are described herein with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It should be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer-readable program instructions.

[0114] These computer-readable program instructions may be provided to a processor of a general-purpose computer, a special-purpose computer, or other programmable data processing apparatus to produce a machine, which, when executed by the processor of the computer or other programmable data processing apparatus, creates a means for implementing the functions / actions specified in one or more blocks of the flowchart and / or block diagram. These computer-readable program instructions may also be stored in a computer-readable storage medium that can direct a computer, a programmable data processing apparatus, and / or other devices to function in a particular manner, such that the computer-readable storage medium having the instructions stored therein includes an article of manufacture that includes instructions for implementing aspects of the functions / actions specified in one or more blocks of the flowchart and / or block diagram.

[0115] Computer-readable program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other devices that can cause a series of operational steps to be performed on a computer, other programmable apparatus, or other devices to produce a computer-implemented process such that the instructions executed on the computer, other programmable apparatus, or other devices implement the functions / actions specified in one or more blocks of the flowchart and / or block diagram.

[0116] The flowcharts and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. For this purpose, each block in the flowchart or block diagram may represent a module, segment, or portion of an instruction, which includes one or more executable instructions for implementing the specified logical function. In some alternative embodiments, the functions noted in the blocks may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending on the functionality involved. It will also be noted that each block of the block diagrams and / or flowchart diagrams, and combinations of blocks in the block diagrams and / or flowchart diagrams, can be implemented by a system based on dedicated hardware that performs the specified functions or actions or a combination of dedicated hardware and computer instructions.

[0117] To provide context for aspects of the disclosed subject matter, Figure 13 and the following discussion is intended to provide a general description of a suitable environment in which aspects of the disclosed subject matter may be implemented. Figure 13 A block diagram of an example non-limiting operating environment that may facilitate one or more embodiments described herein is shown. For brevity, repeated description of similar elements employed in other embodiments described herein is omitted. See Figure 13, a suitable operating environment 1300 for implementing various aspects of the present disclosure may include a computer 1312. The computer 1312 may also include a processing unit 1314, a system memory 1316, and a system bus 1318. The system bus 1318 may operably couple system components including, but not limited to, the system memory 1316 to the processing unit 1314. The processing unit 1314 may be any of a variety of available processors. Dual microprocessor and other multi-processor architectures may also be used as the processing unit 1314. The system bus 1318 may be any of several types of bus structures, including a memory bus or memory controller, a peripheral bus or external bus, and / or a local bus using various available bus architectures, including, but not limited to, Industry Standard Architecture (ISA), Micro Channel Architecture (MSA), Extended ISA (EISA), Intelligent Drive Electronics (IDE), VESA Local Bus (VLB), Peripheral Component Interconnect (PCI), Card Bus, Universal Serial Bus (USB), Advanced Graphics Port (AGP), FireWire, and Small Computer System Interface (SCSI). The system memory 1316 may also include volatile memory 1320 and non-volatile memory 1322. A basic input / output system (BIOS) containing basic routines that transfer information between elements within the computer 1312 during startup may be stored in the non-volatile memory 1322. By way of illustration and not limitation, the non-volatile memory 1322 may include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), flash memory, or non-volatile random access memory (RAM) (e.g., ferroelectric RAM (FeRAM)). The volatile memory 1320 may also include random access memory (RAM) that acts as an external buffer memory. By way of illustration and not limitation, RAM is available in many forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), and Rambus dynamic RAM.

[0118] The computer 1312 may also include removable / non-removable, volatile / non-volatile computer storage media. Figure 13Shows, for example, disk storage 1324. Disk storage 1324 may also include, but is not limited to, devices such as disk drives, floppy disk drives, tape drives, Jaz drives, Zip drives, LS-100 drives, flash cards, or memory sticks. Disk storage 1324 may also include storage media separate from or in combination with other storage media, including but not limited to optical disk drives such as compact disk ROM devices (CD-ROM), CD recordable drives (CD-R drives), CD rewritable drives (CD-RW drives), or digital versatile disk ROM drives (DVD-ROM). To facilitate connection of disk storage 1324 to system bus 1318, a removable or non-removable interface, such as interface 1326, may be used. Figure 13 Also depicted is software that may act as an intermediary between a user and the basic computer resources described in a suitable operating environment 1300. Such software may also include, for example, operating system 1328. The operating system 1328, which may be stored on disk storage 1324, is used to control and allocate the resources of computer 1312. System applications 1330 may utilize the operating system 1328 for resource management through program modules 1332 and program data 1334, such as that stored in system memory 1316 or on disk storage 1324. It should be understood that the present disclosure may be implemented with different operating systems or combinations of operating systems. A user inputs commands or information into computer 1312 through one or more input devices 1336. Input devices 1336 may include, but are not limited to, pointing devices such as mice, trackballs, styli, touch pads, keyboards, microphones, joysticks, game pads, satellite dishes, scanners, TV tuner cards, digital cameras, digital video cameras, web cameras, etc. These and other input devices may be connected to processing unit 1314 via one or more interface ports 1338 through system bus 1318. One or more interface ports 1338 may include, for example, serial ports, parallel ports, game ports, and universal serial bus (USB). One or more output devices 1340 may use some of the same type of ports as input devices 1336. Thus, for example, a USB port may be used to provide input to computer 1312 and output information from computer 1312 to output device 1340. An output adapter 1342 may be provided to account for some output devices 1340 that require special adapters, such as monitors, speakers, and printers, as well as other output devices 1340. By way of illustration and not limitation, output adapter 1342 may include video cards and sound cards that provide connection means between output device 1340 and system bus 1318. It should be noted that other devices and / or systems of devices provide both input and output capabilities, such as one or more remote computers 1344.

[0119] Computer 1312 may operate in a networked environment using a logical connection to one or more remote computers, such as remote computer 1344. Remote computer 1344 can be a computer, server, router, network PC, workstation, microprocessor-based appliance, peer device, or other common network node, etc., and typically may also include many or all of the elements described relative to computer 1312. For brevity, only memory storage device 1346 is shown together with remote computer 1344. Remote computer 1344 can be logically connected to computer 1312 via network interface 1348 and then physically connected via communication connection 1350. Additionally, operations can be distributed across multiple (local and remote) systems. Network interface 1348 can include wired and / or wireless communication networks, such as local area network (LAN), wide area network (WAN), cellular network, etc. LAN technologies include Fiber Distributed Data Interface (FDDI), Copper Distributed Data Interface (CDDI), Ethernet, Token Ring, etc. WAN technologies include, but are not limited to, point-to-point links, circuit-switched networks (such as Integrated Services Digital Network (ISDN) and its variants), packet-switched networks, and Digital Subscriber Line (DSL). One or more communication connections 1350 refer to the hardware / software for connecting network interface 1348 to system bus 1318. Although communication connection 1350 is shown inside computer 1312 for clarity of illustration, it can also be outside computer 1312. The hardware / software for connecting to network interface 1348 can also (for illustrative purposes only) include internal and external technologies, such as modems including conventional telephone-grade modems, cable modems, and DSL modems, ISDN adapters, and Ethernet cards.

[0120] Embodiments of the present invention may be a system, a method, an apparatus, and / or a computer program product at any possible level of integration technology details. The computer program product may include a computer-readable storage medium (or media) having computer-readable program instructions thereon for causing a processor to execute aspects of the present invention. The computer-readable storage medium may be a tangible device that can retain and store instructions for use by an instruction execution device. The computer-readable storage medium may be, for example but not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination of the foregoing. A non-exhaustive list of more specific examples of the computer-readable storage medium may further include the following: a portable computer disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), a static random access memory (SRAM), a portable compact disc read-only memory (CD-ROM), a digital versatile disc (DVD), a memory stick, a floppy disk, a mechanical encoding device (such as punch cards) or raised structures in grooves having instructions recorded thereon), and any suitable combination of the foregoing. As used herein, the computer-readable storage medium should not be construed as a transient signal per se, such as a radio wave or other freely propagating electromagnetic wave, an electromagnetic wave propagating through a waveguide or other transmission medium (e.g., an optical pulse through an optical fiber cable), or an electrical signal transmitted through a wire.

[0121] The computer-readable program instructions described herein can be downloaded to a corresponding computing / processing device from a computer-readable storage medium or downloaded to an external computer or external storage device via a network (e.g., the Internet, a local area network, a wide area network, and / or a wireless network). The network can include copper transmission cables, optical transmission fibers, wireless transmission, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface in each computing / processing device receives the computer-readable program instructions from the network and forwards the computer-readable program instructions for storage in a computer-readable storage medium within the corresponding computing / processing device. The computer-readable program instructions for performing operations for various aspects of the present invention may be assembly instructions, instruction set architecture (ISA) instructions, machine instructions, machine-related instructions, microcode, firmware instructions, state-setting data, configuration data for integrated circuits, or source code or object code written in any combination of one or more programming languages, including object-oriented programming languages such as Smalltalk, C++, etc., and procedural programming languages such as the "C" programming language or similar programming languages. These computer-readable program instructions may be executed entirely on the user's computer, partially on the user's computer, executed as a stand-alone software package, partially on the user's computer and partially on a remote computer, or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network (including a local area network (LAN) or a wide area network (WAN)), or may be connected to an external computer (e.g., through the Internet using an Internet service provider). In some embodiments, an electronic circuit (including, for example, a programmable logic circuit, a field-programmable gate array (FPGA), or a programmable logic array (PLA)) can be customized by utilizing the state information of the computer-readable program instructions to execute the computer-readable program instructions in order to perform aspects of the present invention.

[0122] Aspects of the present invention are described herein with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the invention. It should be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer-readable program instructions. These computer-readable program instructions may be provided to a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions / acts specified in one or more blocks of the flowchart and / or block diagram. These computer-readable program instructions may also be stored in a computer-readable storage medium that can direct a computer, a programmable data processing apparatus, and / or other devices to function in a particular manner, such that the computer-readable storage medium having instructions stored therein comprises an article of manufacture including instructions which implement aspects of the functions / acts specified in one or more blocks of the flowchart and / or block diagram. The computer-readable program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other device to cause a series of operational steps to be performed on the computer, other programmable apparatus or other device to produce a computer-implemented process, such that the instructions which execute on the computer, other programmable apparatus, or other device implement the functions / acts specified in one or more blocks of the flowchart and / or block diagram.

[0123] The flowchart and block diagrams in the figures illustrate the architecture, functionality and operation of possible implementations of systems, methods and computer program products according to different embodiments of the present invention. Accordingly, each block in the flowchart or block diagrams may represent a module, segment, or portion of instructions, which comprises one or more executable instructions for implementing the specified logical function. In some alternative implementations, the functions noted in the blocks may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and / or flowchart illustrations, and combinations of blocks in the block diagrams and / or flowchart illustrations, can be implemented by special-purpose hardware-based systems that perform the specified functions or acts, or combinations of special-purpose hardware and computer instructions.

[0124] Although the subject matter has been described above in the general context of computer-executable instructions of a computer program product running on one computer and / or multiple computers, those skilled in the art will recognize that the present disclosure may also be implemented in conjunction with or in combination with other program modules. Generally, program modules include routines, programs, components, data structures, etc. that perform specific tasks and / or implement specific abstract data types. In addition, those skilled in the art will recognize that the computer-implemented methods of the present invention may be practiced with other computer system configurations, including single-processor or multi-processor computer systems, small computing devices, mainframe computers, and computers, handheld computing devices (e.g., PDAs, telephones), microprocessor-based or programmable consumer or industrial electronic products, etc. The aspects shown may also be practiced in a distributed computing environment where tasks are performed by remote processing devices linked through a communication network. However, some (if not all) aspects of the present disclosure may be practiced on a stand-alone computer. In a distributed computing environment, program modules may be located in both local and remote memory storage devices.

[0125] As used in this application, the terms "component", "system", "platform", "interface", etc. may refer to and / or may include computer-related entities or entities related to an operating machine with one or more specific functions. The entities disclosed herein may be hardware, a combination of hardware and software, software, or software in execution. For example, a component may be, but is not limited to, a process running on a processor, a processor, an object, an executable file, an executing thread, a program, and / or a computer. As an illustration, both an application running on a server and the server may be components. One or more components may reside within a process and / or an executing thread, and a component may be located on one computer and / or distributed between two or more computers. In another example, corresponding components may execute from different computer-readable media having different data structures stored thereon. Components may communicate via local and / or remote procedure calls, such as in accordance with a signal having one or more data packets (e.g., data from a component interacting with another component in a local system, a distributed system, and / or across a network such as the Internet via a signal interacting with other systems). As another example, a component may be a device having a specific function provided by a mechanical component operated by an electrical or electronic circuit, where the electrical or electronic circuit is operated by a software or firmware application executed by a processor. In this case, the processor may be inside or outside the device and may execute at least a portion of the software or firmware application. As yet another example, a component may be a device that provides a specific function through an electronic component without a mechanical component, where the electronic component may include a processor or other means for executing software or firmware, and the software or firmware at least partially imparts the function of the electronic component. In one aspect, a component may emulate an electronic component via, for example, a virtual machine within a cloud computing system.

[0126] In addition, the term "or" is intended to mean an inclusive "or" rather than an exclusive "or". That is, unless otherwise specified or clear from the context, "X employs A or B" is intended to mean any natural inclusive arrangement. That is, if X employs A; X employs B; or X employs both A and B, then "X employs A or B" is satisfied in any of the foregoing instances. Further, the articles "a" and "an" as used in this specification and the drawings generally should be construed to mean "one or more" unless otherwise specified or clear from the context as being directed to the singular form. As used herein, the terms "example" and / or "exemplary" are used to mean serving as an example, instance, or illustration. To avoid doubt, the subject matter disclosed herein is not limited by such examples. Further, any aspect or design described herein as "example" and / or "exemplary" is not necessarily to be construed as preferred or advantageous over other aspects or designs, nor is it intended to exclude equivalent exemplary structures and techniques known to those of ordinary skill in the art.

[0127] As used in this specification, the term "processor" may refer to substantially any computing processing unit or device, including but not limited to a single-core processor; a single processor with software multithreading execution capabilities; a multi-core processor; a multi-core processor with software multithreading execution capabilities; a multi-core processor with hardware multithreading technology; a parallel platform; and a parallel platform with distributed shared memory. Additionally, a processor may refer to an integrated circuit, an application-specific integrated circuit (ASIC), a digital signal processor (DSP), a field-programmable gate array (FPGA), a programmable logic controller (PLC), a complex programmable logic device (CPLD), discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. Further, a processor may utilize nanoscale architectures such as, but not limited to, molecular and quantum-dot based transistors, switches, and gates, in order to optimize space usage or enhance the performance of a user device. A processor may also be implemented as a combination of computing processing units. In this disclosure, terms such as "store", "storage", "data store", "data storage", "database", and substantially any other information storage component related to the operation and functionality of a component are used to refer to a "memory component", an entity embodied in a "memory", or a component that includes a memory. It should be understood that the memory and / or memory components described herein may be volatile memory or non-volatile memory, or may include both volatile memory and non-volatile memory. By way of illustration and not limitation, non-volatile memory may include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable ROM (EEPROM), flash memory, or non-volatile random access memory (RAM) (e.g., ferroelectric RAM (FeRAM)). Volatile memory may include RAM, which may, for example, act as an external cache memory. By way of illustration and not limitation, RAM is available in many forms, such as synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), and Rambus dynamic RAM (RDRAM). Additionally, the memory components of the systems or computer-implemented methods disclosed herein are intended to include, but not be limited to, including these and any other suitable types of memory.

[0128] The foregoing merely includes examples of systems, computer program products, and computer-implemented methods. Of course, for purposes of describing the present disclosure, it is not possible to describe every conceivable combination of components, products, and / or computer-implemented methods, but many further combinations and permutations of the present disclosure will be recognized by those of ordinary skill in the art. Additionally, insofar as the terms “including,” “having,” “possessing,” and the like are used in the detailed description, claims, appendices, and drawings, such terms are intended to be inclusive in a manner similar to the term “comprising,” as “comprising” is construed when used as a transitional word in the claims. The description of the different embodiments has been presented for purposes of illustration, but the description is not intended to be exhaustive or limited to the disclosed embodiments. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope of the described embodiments. The terms used herein were chosen to best explain the principles of the embodiments, the practical application, or a technical improvement found in the marketplace, or to enable those of ordinary skill in the art to understand the embodiments disclosed herein.

Claims

1. A system for topological classification, the system comprising: A memory storing computer-executable components; A classical processor operably coupled to the memory and executing the computer-executable components stored in the memory, wherein the computer-executable components include: At least one quantum computing component operable to encode eigenvalues into the phase of a quantum state of a quantum circuit; and At least one classical computing component that infers one or more Betti numbers using a Bayesian learning algorithm by measuring one or more ancillary states of the quantum circuit, the at least one classical computing component including at least one Bayesian component, wherein the at least one Bayesian component generates one or more Bayesian learning algorithms based on one or more prior knowledge distributions and one or more probability distributions; Wherein the one or more ancillary states include one or more first quantum gates that incorporate one or more variational parameters into the quantum computation, where the variational parameters represent the Bayesian risk parameter and / or the mean of the target distribution in an optimization protocol; Wherein the at least one classical computing component incorporates the prior knowledge into the inference of the one or more Betti numbers via conditional probability; Wherein the at least one classical computing component shares the inferred one or more Betti numbers with the one or more quantum computing components to further refine one or more persistent homology algorithms in subsequent analysis iterations performed by the one or more quantum computing components; and Wherein, in a feedback loop, the computations implemented by the one or more quantum computing components form the basis for the one or more computations implemented by the one or more classical computing components, and vice versa, and the feedback loop iterates until convergence with respect to the inferred Betti numbers and / or until a number of times defined by the user of the system is reached.

2. The system of claim 1, wherein the at least one quantum computing component is operable to encode the eigenvalues of a Laplacian matrix into the phase of a quantum state of a quantum circuit.

3. The system of claim 2, wherein the at least one quantum computing component includes: A subsampling component that selectively samples an object to generate a subsampled object.

4. The system of claim 3, wherein the at least one quantum computing component includes: A homology component that characterizes the subsampled object using a persistent homology algorithm to generate a quantum superposition state.

5. The system of claim 4, wherein the at least one quantum computing component includes: A matrix component that establishes the Laplacian matrix as a quantum object based on the quantum superposition state.

6. The system of claim 5, wherein the at least one classical computing component includes: A measurement component that repeatedly measures the ancillary states and generates a probability distribution, Wherein the at least one classical computing component includes: A sampling component that analyzes the probability distribution using a sampling algorithm to update the prior knowledge distribution.

7. The system according to any one of claims 1-6, wherein the eigenvalue is related to the topological feature of the object.

8. A computer-implemented method for topological classification, the method comprising: encoding an eigenvalue onto a phase of a quantum state of a quantum circuit by the system according to any one of claims 1-7 operably coupled to a processor; and inferring a Betti number by the system using a Bayesian learning algorithm by measuring an ancillary state of the quantum circuit.

9. The computer-implemented method according to claim 8, further comprising: encoding an eigenvalue of a Laplacian matrix onto a phase of a quantum state of a quantum circuit by at least one quantum computing component; and generating a sub-sample object by the at least one quantum computing component by selectively sampling the object.

10. The computer-implemented method according to claim 9, further comprising: characterizing the sub-sample object by the at least one quantum computing component of the system using a persistent homology algorithm to generate a quantum superposition state.

11. The computer-implemented method according to claim 10, further comprising: generating the Laplacian matrix as a quantum object by the at least one quantum computing component of the system based on the quantum superposition state.

12. The computer-implemented method according to claim 11, further comprising: repeatedly measuring the ancillary state by a measurement component of at least one classical computing component of the system, wherein the ancillary state is a product state with the quantum state; and generating a probability distribution by the measurement component of at least one classical computing component of the system based on the repeated measurement.

13. The computer-implemented method according to claim 12, further comprising: analyzing the probability distribution by a sampling component of at least one classical computing component of the system using a sampling algorithm to update a prior knowledge distribution, and generating the Bayesian learning algorithm by a Bayesian component of at least one classical computing component of the system based on the prior knowledge distribution and the probability distribution.

14. The computer-implemented method according to any one of claims 8 to 13, wherein the eigenvalue is related to the topological feature of the object.

15. A computer program product for topological classification, the computer program product comprising: a computer-readable storage medium readable by a processing circuit and storing instructions for execution by the processing circuit to perform the method according to any one of claims 8 to 14.

Images